Mister Exam

Other calculators

The derivative of the function/ e^(2sinx/4)

Derivative of e^(2sinx/4)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 2*sin(x)
 --------
    4    
E
e2sin⁡(x)4e^{\frac{2 \sin{\left(x \right)}}{4}} 
E^((2*sin(x))/4)
Detail solution

  1. Let u=2sin⁡(x)4u = \frac{2 \sin{\left(x \right)}}{4}.

  2. The derivative of eue^{u} is itself.

  3. Then, apply the chain rule. Multiply by ddx2sin⁡(x)4\frac{d}{d x} \frac{2 \sin{\left(x \right)}}{4}:

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. The derivative of sine is cosine:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        So, the result is: 2cos⁡(x)2 \cos{\left(x \right)}

      So, the result is: cos⁡(x)2\frac{\cos{\left(x \right)}}{2}

    The result of the chain rule is:

    e2sin⁡(x)4cos⁡(x)2\frac{e^{\frac{2 \sin{\left(x \right)}}{4}} \cos{\left(x \right)}}{2}

  4. Now simplify:

    esin⁡(x)2cos⁡(x)2\frac{e^{\frac{\sin{\left(x \right)}}{2}} \cos{\left(x \right)}}{2}

The answer is:

esin⁡(x)2cos⁡(x)2\frac{e^{\frac{\sin{\left(x \right)}}{2}} \cos{\left(x \right)}}{2}

The graph
02468-8-6-4-2-10102.5-2.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        2*sin(x)
        --------
           4    
cos(x)*e        
----------------
       2
e2sin⁡(x)4cos⁡(x)2\frac{e^{\frac{2 \sin{\left(x \right)}}{4}} \cos{\left(x \right)}}{2}
Simplify
The second derivative [src] 
                      sin(x)
                      ------
/   2              \    2   
\cos (x) - 2*sin(x)/*e      
----------------------------
             4
(−2sin⁡(x)+cos⁡2(x))esin⁡(x)24\frac{\left(- 2 \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{\frac{\sin{\left(x \right)}}{2}}}{4}
Simplify
The third derivative [src] 
                                  sin(x)
                                  ------
/        2              \           2   
\-4 + cos (x) - 6*sin(x)/*cos(x)*e      
----------------------------------------
                   8
(−6sin⁡(x)+cos⁡2(x)−4)esin⁡(x)2cos⁡(x)8\frac{\left(- 6 \sin{\left(x \right)} + \cos^{2}{\left(x \right)} - 4\right) e^{\frac{\sin{\left(x \right)}}{2}} \cos{\left(x \right)}}{8}
Simplify
    © Mister Exam – Calculator